Why is this true $(-1)^n (-x^{n+1} )= (-x)^{n+1}$?

Question

I found this identity while reading a proof.

The context: $-1\leq x<0$. I am unable to see this identity being true intuitively. Can someone show me a proof of this, or how to get from the left side of the equality to the right side?

Answer 1

$$(-1)^n \cdot -x^{n+1} = (-1)^n \cdot (-1)\cdot x^{n+1} = (-1)^{n+1} x^{n+1} = (-x)^{n+1}$$

Note that $-x^{n+1}$ is not the same as $(-x)^{n+1}$.

Answer 2

$$ (-1)^{n}\cdot (-x^{n+1})=(-1)^{n}\cdot((-1)x^{n+1})=((-1)^{n}(-1))\cdot x^{n+1}=(-1)^{n+1}x^{n+1}=((-1)x)^{n+1}=(-x)^{n+1} $$

Answer 3

we have $$(-1)\cdot (-1)^n\cdot x^{n+1}=(-1)^{n+1}x^{n+1}=(-x)^{n+1}$$